Redefining Network Topology in Complex Systems: Merging Centrality Metrics, Spectral Theory, and Diffusion Dynamics

This paper introduces a novel framework that combines traditional centrality measures with eigenvalue spectra and diffusion processes for a more comprehensive analysis of complex networks. While centrality measures such as degree, closeness, and betweenness have been commonly used to assess nodal importance, they provide limited insight into dynamic network behaviors. By incorporating eigenvalue analysis, which evaluates network robustness and connectivity through spectral properties, and diffusion processes that model information flow, this framework offers a deeper understanding of how networks function under dynamic conditions. Applied to synthetic networks, the approach identifies key nodes not only by centrality but also by their role in diffusion dynamics and vulnerability points, offering a multi-dimensional view that traditional methods alone cannot. This integrated analysis enables a more precise identification of critical nodes and potential weaknesses, with implications for improving network resilience in fields ranging from epidemiology to cybersecurity. Keywords: Centrality measures, eigenvalue spectra, diffusion processes, network analysis, network robustness, information flow, synthetic networks.